Problem about Sum of Squares

[maccaman]

I have a statistics test coming up and we were given two really hard problems to figure out. I don't quite know what they are asking, and we are kinda on our own to solve it. Any help would be greatly appreciated.

1. Given that the equation for the sum of the squares is

SS = x2 - (x)2/n


You are presented with the situation that you have two samples of a variable, each sample of an arbitrary number of observations (in the first situation, assume that the numbers are equal, in the second, assume that the numbers are not necessarily equal). Derive an equation from first principles, or set of equations, that describes the relationship between the variances of the two samples, and the variance of the overall dataset that would exist if the two samples were combined.


Generalise this equation to any arbitrary collection of k different samples (where k is the number of different samples).

2. One of the major problems of analysis is the difference between two samples of a variable. We wish to know if the means of the two samples are different.

Let us imagine that you want to know the mean difference in height between males and females of the same age, by sampling age-matched pairs of otherwise randomly selected people.

Derive an equation that describes the sum of the squares of the difference between two samples.

How different would these equations have to be if the individuals that were sampled were male-female non-identical twins pairs rather than randomly selected people.

 

[Evalesco]

For the first problem, the equation for the sum of the squares of the difference between two samples would be
SS = (x1 - x2)2/(n1 + n2 -2).
In the case of male-female non-identical twin pairs, the equation would be SS = (x1 - x2)2/n.
Hope this helps!

 

[tacman]

The first problem is asking you to derive an equation that describes the relationship between the variances of two samples and the variance of the overall dataset when the two samples are combined. This can be done by first understanding the formula for the sum of squares (SS) which is SS=∑x^2 - (∑x)^2/n. This formula calculates the sum of the squared deviations of each individual data point from the mean of the dataset.

To find the relationship between the variances, we can use the fact that variance is the average of the squared deviations from the mean. So for the overall dataset, the variance would be equal to SS divided by the total number of observations (n). For the first sample, the variance would be SS1 divided by the number of observations in that sample (n1). Similarly, for the second sample, the variance would be SS2 divided by the number of observations in that sample (n2).

When the two samples are combined, the total number of observations would be n1 + n2 and the total sum of squares would be SS1 + SS2. So the variance for the combined dataset would be (SS1 + SS2)/(n1 + n2). By substituting the values of the variances for each sample, we can derive the general equation for the relationship between the variances of two samples and the overall dataset:

Variance of combined dataset = (Variance of sample 1 * Number of observations in sample 1 + Variance of sample 2 * Number of observations in sample 2)/(Total number of observations)

To generalize this equation for any arbitrary collection of k different samples, we can add the variances of each sample and the number of observations in each sample to the equation:

Variance of combined dataset = (Variance of sample 1 * Number of observations in sample 1 + Variance of sample 2 * Number of observations in sample 2 + ... + Variance of sample k * Number of observations in sample k)/(Total number of observations)

The second problem is asking you to derive an equation for the sum of squares of the difference between two samples. This can be done by first finding the difference between the means of the two samples. Let's call this difference d. Then the sum of squares of the difference between the two samples can be calculated as SS = ∑(x - d)^2.

If the individuals sampled were male-female non-identical twin pairs